In this section, our goal is to prove all the theorems in Section .
A crucial part of the proofs is the spectral analysis of the random matrix \(\mathcal {X}\) .
Therefore, we start with introducing the non-asymptotic results of \(\lambda _{\max } (\mathcal {X} \mathcal {X}^\top )\)  and \(\lambda _{\min } (\mathcal {X} \mathcal {X}^\top )\) 
from [1]}.
